The standard method of obtaining invariant factors of a finite abelian group is first to
choose a presentation of the group, and then compute the Smith Normal From (SNF) of the matrix of
relations. For the sandpile group, it is well known that the Laplacian matrix is one of its relation matrices. By computing the SNF of the Laplacian matrix, the sandpile groups for many special families of regular or near regular graphs have been completely or partially determined in the last twenty
years, such as wheel graphs, complete multipartite graphs, Cartesian product of complete graphs \(K_n\square K_m\) , the ladder \(K_2\square C_n\) , the Möbius
ladder graph, a Cayley graph \(D_n\)  of dihedral group, the squared cycle \(C^2_n\) , the thick cycle, etc.; see
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]} and references therein.
